Recently, Schwarz preconditioners – which were already discussed above in the context of GFEM and XFEM [1]}, [2]} – have also gained momentum in immersed finite elements. The concept of Schwarz preconditioning to overcome the problem of linear dependencies on small cut elements was considered in [3]}, [4]}. This concept was generalized to multilevel \(hp\) -finite element bases in [5]}, where it was also demonstrated to be effective in parallel computing frameworks.
In [6]}, Schwarz preconditioning of unfitted systems was applied as a smoother in a multigrid solver, making it suitable for large scale computations. The methodology was applied in high-performance parallel-computing settings in [7]}, [8]}, with [8]} demonstrating excellent scalability for problems with multi-billion degrees of freedom
distributed over close to \(10^5\)  cores. A Balancing Domain Decomposition by Constraints (BDDC) scalable preconditioner, tailored to immersed FEM by choosing appropriate weighting coefficients for cut basis functions, has been proposed in [10]}. BDDC methods are multilevel additive domain decomposition algorithms that can scale up to millions of cores/subdomains [11]}. The preconditioner in [10]} results in an effective preconditioner for linear basis functions and exhibits the same parallelism potential as standard BDDC, thus being well-suited for large-scale systems on distributed-memory machines.
